Algebraic methods in random matrices and enumerative geometry

Bertrand Eynard (1) , Nicolas Orantin (2)

1 SPhT - Service de Physique Théorique
2 CERN - European Organization for Nuclear Research

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg , and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...

Mots clés: random matrices; algebraic geometry; topological strings; integrability

Domaines: Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]

Bertrand Eynard, Nicolas Orantin. Algebraic methods in random matrices and enumerative geometry. 2008. ⟨hal-00340602⟩